Textured pattern

Where r represents the distances on the pattern, we must determine constants A,B and y of the two-dimensional lattice (Fig. 8) In real situations, reflections on texture patterns are in the sharp of arcs. This, may be explained as follows. The intersection of the rings of the reciprocal lattice with a plane gives, in the ideal case, a point. However with real textures, because of a certain disorder in the crystal orientation relative to the texture... 

Figure 11. (a) formation of layer lines in the reciprocal lattice of a texture for a orthogonal unit cell, (b) the doubling of number of circular scattering regions in the reciprocal lattice of a texture and therefore the number of reflections on an ellipse of a pattern for a non-orthogonal unit cell, (c) measurement of a values of 2r and 2 D on a texture pattern. 

Key words texture patterns, electron diffraction, intensities extraction. 

Abstract The program TexPat was developed for quantification of texture patterns in... 

Many compounds, including clay minerals, form needle- or plateshaped crystals. With finely dispersed minerals, the electron diffraction method can give a special kind of diffraction pattern, the texture pattern, which contains a two dimensional distribution of a regularly arranged set of 3D reflections [2], Specimens of fine-grained lamellar or fiber minerals, prepared by sedimentation from suspensions onto supporting surfaces or films, form textures in which the component microcrystals have a preferred orientation. Texture patterns of lamellar crystals tilted with respect to the electron beam are called oblique texture electron diffraction patterns [1]. 

However, quite complicated algorithms are needed for extracting the data Ifom texture patterns. The analysis of texture patterns must be performed in a different way compared to regular electron diffraction patterns, due to different geometrical settings. Firstly, the centre of the... 

The purpose of this chapter is to describe the basic geometrical properties of texture patterns, which lead to an understanding of possible methods needed for extracting the quantitative information from diffraction patterns of this kind. 

Figure 1. Electron diffraction pattern examples a - spot pattern from a single crystal, b -oblique texture pattern tilted by 60°, c - powder ring pattern.

Figure 2. Fonnation of ring and oblique texture patterns, a - several randomly rotated artificial crystallites and its Fourier transform (inset) b - reciprocal space with rings and zero tilt Ewald sphere c - 60° tilt of the Ewald sphere (reflection centers lie on the ellipse) d - the diffraction pattern as it is seen on the image plane.

Oblique texture patterns have almost perfect 2mm symmetry and thus the whole set of diffraction spots is represented by the reflections in one quadrant. The arcs are exactly symmetrically placed relative to the major axis, being sections of the same spherical band in reciprocal space. The reflections on the lower half of the pattern are sections of reciprocal lattice rings, which are Friedel partners and thus equivalent to those giving the reflections of the upper half assuming a flat surface of the Ewald sphere. Actually, if the curvature of the Ewald sphere is taken into account, the upper and lower parts of a texture pattern will differ slightly. 

The purpose of indexing texture patterns is the geometrical reconstruction of the three-dimensional reciprocal lattice from the two-dimensional distribution of H spacings. One advantage of texture patterns is the possibility to determine all unit cell parameters of a crystal unambiguously and index all the diffraction peaks from only a single texture... 

The indexing is based on the direct relationship between the texture pattern and the reciprocal lattice. For a reflection hkl with the reciprocal lattice xector Hhki. ... 

Indexing a texture pattern is often rather complicated, but several characteristic features facilitate the indexation (see [142]). 

Since diffraction pattern has a scale factor LI all distances measured on the pattern should be recalculated into reciprocal. Following Zvyagin s notation one measures B k values on the texture pattern, which correspond to reciprocal values Hgoh by... 

Figure 4. Indexing texture patterns from crystals of different symmetries (see the text for explanations). Assigning the indices to the reflections is shown in (a) - (c). The splitting of the reflections hkl for different symmetries is shown in (d) - (f). (a), (d) trigonal (lizardite IT), (b), (e) monoclinic (muscovite 2Mi). (c), (f) triclinic (kaolinite).

There are two methods for texture pattern intensity measurements described in [3]. The first approach is to perform integration of a small region at the centre of the arc-shaped reflection. It can be done by hand and has been used by Vainshtein, Zvyagin and others. 

The lack of software for the decomposition method in two dimensions in application to electron diffraction texture patterns was avoided by Zvyagin and Zhukhlistov by using one-dimensional methods of intensity extraction. Later the decomposition method for one-dimensional radial profiles (starting at the centre of the diffraction pattern and passing through... 

For oblique texture patterns, when the tilting angle (p is taken into account we have (in relative values) ... 

In the case of texture patterns none of the steps is trivial. 

Here we will describe an example of data extraction during processing of texture pattern of brucite and show the results of indexing, determination of unit cell parameters and extraction of intensities. 

The first critical step in the analysis of any texture pattern is to determine the position of the centre (000 reflection), since all the other steps are dependent on a correctly placed centre. The centre can be positioned automatically or manually. 

Alternatively, the user can choose four S5mimetry related reflections, with the same d-value, to determine the centre from those points (four-clicks method). Then the centre of the texture pattern is at the centre of the circle containing those four reflections (see Fig. 5a). 

The difference between automatically calculated centre position and the centre calculated from the user-defined four reflections was found to be -Yz pixel in each x- and j-directions on texture patterns with the size of 1024x1280 pixels. 

Summarizing the results we can make a conclusion that all the methods of the centre calculation are almost equal in the precision. The maximum difference between the results obtained using described methods was 2 pixels, which is less than 0.003 A (the scale factor was about 750 to 900 pixels/A) for the given texture patterns. Such small difference has no significant influence on further calculations, which was proved by calculations performed on these texture patterns. 

In the texture pattern described in this chapter, the background was approximated by a Gaussian curve from three different radial profiles, since there can be some deviations from the average radial profile in different directions which can lead to over- or underestimations of the background. Each profile was averaged over 3° in the azimuthal direction the directions... 

Figure 6. Background estimation of a texture pattern (a) before and (b) after background subtraction.

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